What is the vector cross product in an oblique coordinate system?

[KleZMeR]

Homework Statement

Find vector product of $C = A \times B$ of two vectors in oblique coord. system. Give explicit expressions of components of C in covariant and contravariant components (constructing reciprocal basis from direct basis will be useful).

Homework Equations

I am basically just crossing two vectors, one product is that of two arbitrary contravariant vectors, and one is a product of two arbitrary covariant vectors. I understand this part, I just am always confused by the word "explicit."

The Attempt at a Solution

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I take this determinant (contravariant)
$\begin{array}{ccc} a_1 & a_2 & a_3 \\ A^1 & A^2 & A^3 \\ B^1 & B^2 & B^3 \end{array}$

and this one as well (covariant)
$\begin{array}{ccc} a^1 & a^2 & a^3 \\ A_1 & A_2 & A_3 \\ B_1 & B_2 & B_3 \end{array}$

and for my covariant components I can set $a_i = \frac{e_i}{sin(\alpha)}$

I am not sure what else is being asked, if anything at all.

I have this relation for components:

$x_1 = x^1 + x^2cos(\alpha)$

but not sure if I should apply it to get vector terms as $a_1A_1 +...$:

 

[jedishrfu]

I found this via Google and thought it may help you resolve your problem:

http://www.mc.maricopa.edu/~kev2077170/i256/Nonortho_math.pdf

I couldn't find any other reference on vector cross products in oblique 3D coordinates except for this book reference:

http://books.google.com/books?id=gJ...v=onepage&q=oblique coordinate system&f=false